It will take 6-1 months for the full recovery of the nail bed and nail. Hydrogen peroxide isn't really used much by modern medical prectitioners so you may want to ditch it for future used. A good washing with soap and water and antibiotic ointment & bandaid would have been just perfect.
But from experience I can tell you it will be nearly a year before her nail will begin to look normal. But it should become normal eventually. The other option is to see a hand specialist or, believe it or not, a foor doctor since they are very involved with nail care. (It may not be covered under insurance since it's a foot doc caring for a hand).
Good luck and do not beat your self uop over this - it's a finger nail not cancer. We mom are way too quick to take on unmerited guilt!